Solution of Vacuum Field Equation Based on Physics Metrics in Finsler Geometry and Kretschmann Scalar

Archive of SID

Int. J. Nonlinear Anal. Appl. 10 (2019) Special Issue (Nonlinear Analysis in Engineering and Sciences), 97-114 ISSN: 2008-6822 (electronic) https://dx.doi.org/10.22075/IJNAA.2019.4403 Solution of Vacuum Field Equation Based on Physics Metrics in Finsler Geometry and Kretschmann Scalar

M. Farahmandy Motlagh∗, A. Behzadi Department of Mathematics, Faculty of Basic Sciences, University of Mazandaran, P. O. Box 47416-95447, Babolsar, Iran.

Abstract The Lemaître-Tolman-Bondi (LTB) model represents an inhomogeneous spherically symmetric universe filled with freely falling dust like matter without pressure. First, we have considered a Finslerian anstaz of (LTB) and have found a Finslerian exact solution of vacuum field equation. We have obtained the R(t, r) and S(t, r) with considering establish a new solution of Rµν = 0. Moreover, we attempt to use Finsler geometry as the geometry of spacetime which compute the Kretschmann scalar. An important problem in General Relativity is singularities. The curvature singularities is a point when the scalar curvature blows up diverges. Thus we have determined Ks singularity is at R = 0. Our result is the same as Reimannian geometry. We have completed with a brief example of how these solutions can be applied. Second, we have some notes about anstaz of the Schwarzschild and Friedmann- Robertson- Walker (FRW ) metrics. We have supposed condition d log(F ) = d log(F¯) and we have obtained F¯ is constant along its geodesic and geodesic of F . Moreover we have com- 2 puted Weyl and Douglas tensors for F and have concluded that Rijk = 0 and this conclude that 2 2 Wijk = 0, thus F is the Ads Schwarzschild Finsler metric and therefore F is conformally flat. We have provided a Finslerian extention of Friedmann- Lemaitre- Robertson- Walker metric based on solution of the geodesic equation. Since the vacuum field equation in Finsler spacetime is equivalent to the vanishing of the Ricci scalar, we have obtained the energy- momentum tensor is zero. Keywords: Einstein’s equations, Lemaître–Tolman–Bondi; Kretschmann scalar, Finsler Geometry, Friedmann-Robertson-Walker, Schwarzschild.

1. Introduction Cosmic structures today have entered the non-linear structure. They can not on all scales be described by a linear perturbation theory on top of the Friedmann-Lemaitre-Robertson-Walker (FLRW) metric

∗Corresponding author Email address: [email protected], [email protected] (M. Farahmandy Motlagh∗, A. Behzadi) Received: July 2019 Revised: October 2019 www.SID.ir 98Archive of SID M. Farahmandy Motlagh, A. Behzadi

[1]. Lemaitre-Tolman-Bondi (LTB) solutions are used frequently to describe the collapse or expansion of spherically symmetric inhomogeneous mass distributions in the Universe. The LTB solutions contain as special cases both the Schwazchild and dust FLRW solutions. The LTB metric is the simplest spherically symmetric solution of Einstein equations representing an inhomogeneous dust distribution. It may be written in synchronous coordinates as [2]:

′ 2 R F 2 = −ytyt + ( )yryr + R2(r, t)(yθyθ + sinθyφyφ), 1 + E The Kretschmann scalar gives the amount of curvature of spacetime, as a function of position near (and within) a black hole [3]. In many cases one of the most useful ways to check that is by checking for the finiteness of the Kretschmann scalar. The alternative way to describe the black holeswithin the cosmological surrounding is to use the Lemaitre-Tolman-Bondi (LTB) models for inhomogeneous matter distribution that was fundamentally explored in [4]. There are very few exact solutions of the Einstein equations, but perhaps the most well-known solution was first derived by Schwarzschild. Exact solutions of Einstein’s field equations have played important roles in the discussion of physical problems. Obvious examples are the Schwarzschild and Kerr solutions for studying black holes and the Friedman solutions for cosmology [5]. Gravitational field equations describing the geometry of space-time play a fundamental role inmod- ern theoretical physics. There are many methods in mathematical physics for studying such systems of equations. For example, one can use methods of additional symmetries of the system of equations, the Hamiltonian formulation of the theory of dynamical systems, etc. Essentially, all physically interesting cases (FRW cosmology, black hole) belong to Stackel and homogeneous spaces [6] In contrast, the cosmic fluid in the FRW model is supposed to possess arbitrary high pressure apart from arbitrary high density. But there is no pressure gradient to support the fluid against its self- gravity [7]. The paper is organized as follows. We review some basic matterial of Finsler geometry and list of tools that is needed in context. In section III we establish a new solution of Rµν = 0 which appears curious and explains the source of curvature of the well-known Lemaître–Tolman–Bondi(LTB) solution. Moreover, we have analytic solution to the geodesic equations. In the section V we compute Kretschmann scalar and show that the LTB singularity is at R = 0. In the section VI we obtain Kretschmann scalar for a one example. The section VII presents opinion of Li and Chang [8] to other words. We consider an ansatz of the Schwarzschild metric F 2 and we show results of vacuum solution. In the section VIII, we compute Weyl and Douglas tensors for F 2 and conclude that 2 Rijk = 0 and this conclude that Wijk = 0, thus F is the Ads Schwarzschild Finsler metric and therefore F 2 is conformally flat. Then we consider Douglas tensor if D = 0 we conclude that F 2 is the Schwarzschild metric. In the section IX, we consider F LRW metric that it has the Finslerian structure then if only we suppose that the radial motion of particles, and notice the velocity of a dr particle is small, therefore the energy-momentum tensor is zero. dt

2. Preliminaries We begin with a very brief discussion of Finsler geometry. A Finsler metric is a continuous function F : TM → [0, ∞] with the following properties. (1) Regularity: F is smooth on TM \ 0 := {(x, y) ∈ TM|y ≠ 0}. (2) Positive homogeneity: F (x, λy) = λF (x, y) for all λ > 0.

www.SID.ir Solution of Vacuum Field Equation Based on Physics Metrics in Finsler ... 10Archive (2019) Special of SID Issue (Nonlinear Analysis in Engineering and Sciences), 97-114 99

∂ ∂ 1 2 (3) Strong convexity: the fundamental tensor gij := ∂yi ∂yj ( 2 F ). is positive definite at all (x, y) ∈ TM \ 0 [9]. A positive, zero and negative F correspond to time-like, null and space-like curves, respectively. For a Finsler metric F = F (x, y) its geodesics are characterized by the system of differential equations

d2xµ + 2Gµ = 0, (2.1) dτ 2 Where the local functions Gµ = Gµ(x, y) are called the spray coeﬀicients and are given by

1 ∂2F 2 ∂F 2 Gµ = gµν( yλ − ), (2.2) 4 ∂xλ∂yν ∂xν

µν −1 Where y ∈ TxM and (g ) := (gµν) . The Cartan tensor quantifies, the deviation of F from being Riemannian, and is defined as ∂ ∂ ∂ 1 A := ( F 2), (2.3) ijk ∂yi ∂yj ∂yk 4

Where the components Aijk are minus one-homogeneous symmetric (0, 3)-tensor field. In order to calculate the Kretschmamm scalar, we need first to calculate the Christoffel symbols by differentiating the metric of F . 1 δg δg δg Γµ = gµl( lσ + lν − σν ), (2.4) νσ 2 δxν δxσ δxl δ ∂ − ∂Gρ ∂ where δxµ = ∂xµ ∂yµ ∂yρ . In Finsler geometry once the Christoffel symbols are calculated then we calculate the Riemann curvature of Chern connection to be: δΓµ δΓµ Rµ = νλ − νσ + Γl Γµ − Γl Γµ , (2.5) νσλ δxσ δxλ νλ lσ νλ lλ

λ From the Riemann tensor R νρσ one can calculate the Riemann curvature tensor Rµνρσ as follows:

λ Rµνρσ = gµλR νρσ, (2.6)

The inverse of Rµνρσ can be compute by

µνρσ µi νj ρk σl R = g g g g Rijkl, (2.7)

The Kretschmann invariant is µνρσ Ks = R Rµνρσ (2.8) Because it is a sum of squares of tensor components, this is a quadratic invariant. For the use of a computer algebra system a more detailed writing is meaningful:

µi νj ρk σl Ks = g g g g RijklRµνρσ, (2.9)

The Ricci tensor Rµν, it is the Riemann curvature tensor with the first and third indices contracted,

αβ β Rµν = g Rαµβν = Rµβν, Which is a symmetric tensor, Rµν = Rνµ,

www.SID.ir 100Archive of SID M. Farahmandy Motlagh, A. Behzadi

It is straightforward to show that, because of the symmetry relations, the alternative contraction leads to the same Ricci tensor αβ Rµν = −g Rµαβν, The Ricci scalar R, it is the Riemann curvature tensor contracted twice.

αβ β R = g Rαβ = Rβ, By definition, the Ricci scalar is a positively homogeneous function of degree twoin y ∈ TM. For a Finsler metric F , the S−curvature is given by following:

µν S = g Ricµν, (2.10) and, as a consequence, the modified Einstein tensor in Finsler space time can be obtained as 1 G ≡ Ric − g S, (2.11) µν µν 2 µν µ The covariant derivative of Gν in Finsler spacetime is given as δ Gµ = Gµ + Γµ Gρ − Γρ Gµ, (2.12) ν|k δxk ν kρ ν kν ρ µ ′|′ Γkρ is the Chern connection and to denote the covariant derivative. Here, we consider also the Rici scalar which is expressed entirely in terms of x and y derivatives of spray coeﬀicients Gµ as follows: ∂Gµ ∂2Gµ ∂2Gµ ∂Gµ ∂Gλ RicF 2 = 2 − yλ + 2Gλ − , (2.13) ∂xµ ∂xλ∂yµ ∂yλ∂yµ ∂yλ ∂yµ

3. Vacuum solution of Finsler metric by using Ansatz of the LTB The theory of relativity describes the relation between the curvature of spacetime to the energy of an object. The vacuum field equations of General Relativity are Rµν = 0. Exact solutions of Einstein’s field equations have played important roles in the discussion of physical problems. Ob- vious examples are the Schwarzschild and Kerr solutions for studying black hole and the Friedman solutions for cosmology. Since dust is believed to be an appropriate representation of the universe’s matter content on the large scale at the present time, LTB solutions have been much used to provide exact models of structures in the universe [10]. First of all we consider an Finslerian ansatz in the form of bellow then we obtain the solutions are similar to the LTB solutions in general relativity.

F 2 = −ytyt + S2(r, t)yryr + R2(r, t)F¯2(θ, φ, yθ, yφ), (3.1) We begin by calculating the connection symbols based on the spherically symmetric form of (3.1). It will be convenient to introduce the notation and we obtain some results, where the Finsler metric can be derived as 2 2 2 gµν = diag(−1,S ,R g¯θθ,R g¯φφ), (3.2) 1 1 1 gµν = diag(−1, , g¯θθ, g¯φφ), (3.3) S2 R2 R2 µν −1 ¯ij The matrix gµν is invertible and its inverse is denoted by g = [gµν] and g¯ij and g are components of the metric derived from F¯ and the indices i,j run over the angular coordinates θ,φ . plugging the Finsler structure (3.1) into (2.2), we obtain that SS˙ RR˙ Gt = yryr + F¯2, (3.4) 2 2 www.SID.ir Solution of Vacuum Field Equation Based on Physics Metrics in Finsler ... 10Archive (2019) Special of SID Issue (Nonlinear Analysis in Engineering and Sciences), 97-114 101

′ ′ S S˙ RR Gr = yryr + yryt − F¯2, (3.5) 2S S 2S2 ′ R˙ R Gθ = ytyθ + yryθ + G¯θ, (3.6) R R ′ R˙ R Gφ = ytyφ + yryφ + G¯φ, (3.7) R R Here dot and prime denote partial derivatives with respect to the parameters t and r respectively. G¯ are the geodic spray coeﬀicient derives from F¯. Plugging the geodesic coeﬀicients(3.4) − (3.7) into the formula for the Ricci scalar (2.13), we obtain that

′′ −2R¨ S¨ −2R 2R˙ SS˙ RicF 2 = RµF 2 = ( − )ytyt + ( + µ R S R R ′ ′ ′ ′ 2R S 4R˙ 4SR˙ + + SS¨)yryr + (− + )yryt + (R˙ 2 RS R RS ′ 2 ′ ′ ′′ R S˙RR˙ R S R R R − + RR¨ + + − + Ric¯ )F¯2, S2 S S3 S2 (3.8) where Ric¯ denotes the Ricci scalar of the Finsler structure F¯. Since the vacuum field equation in Finsler spacetime is equivalent to the vanishing of the Ricci scalar and F¯ is independent of yt and yr, the vanishing of Ricci scalar implies that the terms in each square bracket of (3.8) should vanish as well. These equations are given as −2R¨ S¨ − = 0, (3.9) R S ′′ ′ ′ −2R 2R˙ SS˙ 2R S + + + SS¨ = 0, (3.10) R R RS ′ ′ −4R˙ 4SR˙ + = 0, (3.11) R RS ′ 2 ′ ′ ′′ R S˙RR˙ R S R R R R˙ 2 − + RR¨ + + − + Ric¯ = 0, (3.12) S2 S S3 S2 From equation (3.11) we obtain that ′ ′ −4R˙ −4SR˙ = , (3.13) R RS We remove the same variable from both sides of equation (3.13) at the same time, we obtain

′ ′ R˙ S = SR˙ , (3.14)

′ R˙ S˙ ′ = , (3.15) ∫ R S dx Integrating this and using the formula x = ln x we get,

∫ ′ ∫ R˙ S˙ dt = dt, (3.16) R′ S

www.SID.ir 102Archive of SID M. Farahmandy Motlagh, A. Behzadi

Upon evaluating each of these integrals we should get a constant of integration for each integral since we really are doing two integrals. Since there is no reason to think that the constants of integration will be the same from each integral we use different constants for each integral. Now, both c and k are unknown constants and so the sum of two unknown constants is just an unknown constant and we√ acknowledge that by simply writing the sum as a c. Moreover, c is arbitrary, we can choose c = ln( 1 + E). So, the integral is then,

′ √ ln R = ln S + ln( 1 + E), (3.17)

Apply the product rule for logarithms, we have

′ √ ln R = ln S( 1 + E), (3.18)

After simplifying equation (3.18), we get

′ R √ = S, (3.19) 1 + E Equations (3.1) and (3.19) show that F 2 is similar to the LTB solution

′ 2 R F 2 = −ytyt + ( )yryr + R2(r, t)F¯2(θ, φ, yθ, yφ) (3.20) 1 + E Noticing that Ric¯ is independent of r and t, thus equation (3.12) is satisfied if and only if Ric¯ is constant. This means that the two- dimensional Finsler space F¯ is constant flag curvature space. Therefore Ric¯ = λ. From equation (3.9), we obtain

−2R¨ S¨ = , (3.21) R S From simplifying equation (3.21), we obtain

−SR¨ R¨ = , (3.22) 2S

R2 With using equation (3.10) and multiplying the two sides of equation by 2S2 , we have

′′ ′ ′ −RR R˙ SR˙ RR S −R2S¨ + + = , (3.23) S2 S S3 2S By substituting equation (3.22) into equation (3.23), we get

′′ ′ ′ −RR R˙ SR˙ RR S + + = RR,¨ (3.24) S2 S S3 Equations (3.12) and (3.24) conclude that

′ 2 R R˙ 2 − + 2RR¨ + λ = 0, (3.25) S2 By plugging equation (3.19) into equation (3.25) we get following equations

′ 2 R (1 + E) R˙ 2 − + 2RR¨ + λ = 0, (3.26) R′ 2 www.SID.ir Solution of Vacuum Field Equation Based on Physics Metrics in Finsler ... 10Archive (2019) Special of SID Issue (Nonlinear Analysis in Engineering and Sciences), 97-114 103

R˙ 2 − (1 + E) + 2RR¨ + λ = 0, (3.27) R˙ 2 + 2RR¨ = E, (3.28) Therefore Here angular distance R, depending on the value of t and E(r), is given by √ R(t, r) = E(r)t, (3.29)

By considering λ = 1, (3.30) Thus our LTB metric is,

′ 2 E t2 F 2 = −ytyt + yryr + E(r)t2F¯2(θ, φ, yθ, yφ), (3.31) 4E(1 + E)

4. Analytic solution to the geodesic equations of the finsler geometry In this section, for studying the geodesics of the test particles in the finsler geometry, the equations for the geodesic sprays coeﬀicients (3.4) − (3.7) plug into the geodesic equation (2.1), we obtain the geodesic equation of Finsler spacetime (3.1),

d2t SS˙ RR˙ + yryr + F¯2 = 0, (4.1) dτ 2 2 2

′ ′ d2r S S˙ RR + yryr + yryt − F¯2 = 0, (4.2) dτ 2 2S S 2S2 ′ d2θ R˙ R + ytyθ + yryθ + G¯θ = 0, (4.3) dτ 2 R R ′ d2φ R˙ R + ytyφ + yryφ + G¯φ = 0, (4.4) dτ 2 R R Noticing that F¯ is independent of r and t, thus equations (4.1) and (4.2) are satisfied if and only if F¯ is constant along the geodesic.

5. Kretschmann Scalar What is the nature of the singularity ?

Definition 1. The point z is called a singular point or singularity of K if K is not analytic at z but every neighborhood of z contains at least one point at which K is analytic.

The Kretschmann scalar gives the amount of curvature of spacetime, as a function of position near (and within) a black hole. [1]. The derivation of the Kretschmann scalar is simple in principle, but requires tedious algebraic computation in practice. From the specified metric, we compute, first, the connection, which is not itself a tensor. Next, we compute the Riemann tensor and Kretschmann scalar components. The non-vanishing Christoffel symbols for the metric (3.1) are given by

′ S˙ S˙ S Γt = SS,˙ Γt = RR˙ g¯ , Γt = RR˙ g¯ , Γr = , Γr = , Γr = rr θθ θθ φφ φφ tr S rt S rr S

www.SID.ir 104Archive of SID M. Farahmandy Motlagh, A. Behzadi

′ ′ ′ ′ −R R −R R R˙ R R R˙ Γr = g¯ , Γr = g¯ , Γθ = , Γθ = , Γθ = , Γθ = , θθ S2 θθ φφ S2 φφ tθ R rθ R θr R θt R 1 ∂g¯ 1 ∂g¯ 1 ∂g¯ −1 ∂g¯ Γθ = g¯θθ θθ Γθ = g¯θθ θθ , Γθ = g¯θθ θθ Γθ = g¯θθ φφ θθ 2 ∂θ θφ 2 ∂φ φθ 2 ∂φ φφ 2 ∂θ ′ ′ R˙ R R R˙ −1 ∂g¯ 1 ∂g¯ Γφ = , Γφ = , Γφ = , Γφ = , Γφ = g¯φφ θθ Γφ = g¯φφ φφ tφ R rφ R φr R φt R θθ 2 ∂φ θφ 2 ∂θ 1 ∂g¯ 1 ∂g¯ Γφ = g¯φφ φφ Γφ = g¯φφ φφ , (5.1) φθ 2 ∂θ φφ 2 ∂φ With the connection in hand, we are in a position to compute the Riemann tensor itself:

t t Rtνρσ = gttR νρσ = (−1)R νρσ, (5.2)

r 2 r Rrνρσ = grrR νρσ = S R νρσ, (5.3) θ 2 θ Rθνρσ = gθθR νρσ = (R g¯θθ)R νρσ, (5.4) φ 2 φ Rφνρσ = gφφR νρσ = (R g¯φφ)R νρσ, (5.5) We use maple program and manual computing and obtain a second scalar i.e: Kretschmann scalar as follows:

′ 20(E(r))2 − 9((E(r)) )2 + 20E(r) K = s 2R4 1 + g¯θθg¯φφ(∂ g¯θθ∂ g¯ − ∂ g¯θθ∂ g¯ )2 2R4 θ φ θθ φ θ θθ 1 + g¯θθg¯φφ(∂ g¯φφ∂ g¯ − ∂ g¯φφ∂ g¯ )2 2R4 θ φ φφ φ θ φφ 2 2 + (¯gφφ)2(Rθ )2 + (¯gθθ)2(Rφ )2, R4 φθφ R4 θφθ (5.6)

Therefore the LTB solution has singularity at R = 0.

6. example ¯ For example, we make subspace F to be a “Finslerian sphere” FFS. Then, the exterior metric of vacuum field solution was given as We finished with a brief example of how these solutions canbe applied. Now, we consider the Finsler structure in the form

F 2 = −ytyt + S2(r, t)yryr √ + R2(r, t)( yθyθ + yφyφ − sin θyφ)2, (6.1)

Where √ F¯2 = yθyθ + yφyφ + sin2 θyφ − 2 yθyθ + yφyφ sin θyφ, − φ 3 2 sin θ(y ) g¯θθ = R (t, r)( 3 + 1), (yθyθ + yφyφ) 2

www.SID.ir Solution of Vacuum Field Equation Based on Physics Metrics in Finsler ... 10Archive (2019) Special of SID Issue (Nonlinear Analysis in Engineering and Sciences), 97-114 105

θ θ φ φ 3 θθ (y y + y y ) 2 g¯ = 3 , R2(− sin θ(yφ)3 + (yθyθ + yφyφ) 2 ) 2 2 θ 2 φ 2 3 φ φ 3 θ 2 R ((2 − cos θ)((y ) + (y ) ) 2 − sin θy (2(y ) + 3(y ) )) g¯φφ = 3 , (yθyθ + yφyφ) 2 3 − θ θ φ φ 2 φφ (y y + y y ) g¯ = 3 , R2((cos2 θ − 2)(yθyθ + yφyφ) 2 + sin θyφ(2(yφ)3 + 3(yθ)2)) F 2 is Finslerian metric, since the fundamental form g¯ is function of (θ, φ, yθ, yφ). Therefore, we can compute the Kretschman scalar by using the equation (5.6). It is convenient to consider the orbit of π particle confined to the equatorial plane θ = 2 . ′ 20(E(r))2 − 9((E(r)) )2 + 20E(r) K = s 2R4 Q + 3 , (−((yθ)2 + (yφ)2) 2 + (yφ)3)2R8 (6.2) Where − ′2 θ 2 φ 2 3 1 2 R 2 Q =(4(−((y ) + (y ) ) 2 ( + R ( − R˙ ))) 2 S2 ′2 2 φ 3 R 2 2 4 φ 3 θ 2 φ 2 3 + R (y ) ( − R˙ )) (−R (y ) ((y ) + (y ) ) 2 S2 1 (R4 + 1)(yφ)2 + (R4 + )(yφ)6 + (3(yφ)4 2 2 + 3(yφ)2(yθ)2 + (yθ)4), (6.3) Which represents that the curvature singularity is located at R = 0.

7. Vacuum solution of Finsler metric of the Schwarzschild metric First of all we consider an ansatz of the Schwarzschild metric and we show results of vacuum solution are different from of [8]. Then, we consider FRW metric and compute the geodesic equations. We consider an ansatz of the Schwarzschild metric which was considered in [8] of the form F 2 = B(r)ytyt − A(r)yryr − r2F¯2(θ, φ, yθ, yφ), (7.1) and we obtain some results, where the Finsler metric can be derived as

2 gµν = diag(B, −A, −r g¯ij), (7.2) gµν = diag(B−1, −A−1, −r−2g¯ij), (7.3) ¯ij ¯ where g¯ij and g are components of the metric derived from F . Plugging the Finsler structure (7.1) into equation (2.2), we obtain

′ B Gt = ytyr, (7.4) 2B www.SID.ir 106Archive of SID M. Farahmandy Motlagh, A. Behzadi

′ ′ A B r Gr = yryr + ytyt − F¯2, (7.5) 4A 4B 2A 1 Gθ = G¯θ + yθyr, (7.6) r and 1 Gφ = G¯φ + yφyr. (7.7) r We consider variations of F¯ along the geodesic, therefore we have dF¯ δF¯ dxi ∂F¯ δyi = + F¯ , (7.8) dτ¯ δxi dτ¯ ∂yi F¯ dτ¯ By simplifying equation (7.8) as following,

i i j i i 1 ∂G j dF¯ ∂F¯ dx 1 ∂G¯ ∂F¯ dx ∂F¯ dy + j dx = − + F¯ ( 2 ∂y ) dτ¯ ∂xi dτ¯ 2 ∂yi ∂yj dτ¯ ∂yi F¯ dτ¯ ∂F¯ dxi 1 ∂G¯j ∂F¯ dxi ∂F¯ dyi 1 ∂F¯ ∂G¯i dxj = − + + ∂xi dτ¯ 2 ∂yi ∂yj dτ¯ ∂yi dτ¯ 2 ∂yi ∂yj dτ¯ ∂F¯ dxi ∂F¯ dyi = + , (7.9) ∂xi dτ¯ ∂yi dτ¯ where 1 ∂2F¯2 ∂F¯2 G¯ = ( yk − ), (7.10) i 4 ∂xk∂yi ∂xi by multiplying yi in the above equation, we obtain the following equations, 1 ∂F¯ yiG¯ = F¯ yi, (7.11) i 2 ∂xi and 2yiG¯ ∂F¯ i = yi, (7.12) F¯ ∂xi as we know, dF¯ ∂F¯ dyi = , (7.13) dτ¯ ∂yi dτ¯ and ∂F¯ d2xi ∂F¯ ∂F¯ ∂F¯ = (−2G¯θ) + (−2G¯φ) = −2 G¯i ∂yi dτ¯2 ∂yθ ∂yφ ∂yi ∂F¯ ∂F¯ yiyj G¯ = −2 g¯ijG¯ = −2 G¯ = −2yj j . (7.14) ∂yi j ∂yi F¯2 j F¯ By using the equations (7.12) and (7.14), one rewrite the equation (7.9) as, dF¯ = 0. (7.15) dτ¯ Therefore F¯ is constant along its geodesic. We know that dF¯ dF¯ dτ¯ = = 0, (7.16) dτ dτ¯ dτ www.SID.ir Solution of Vacuum Field Equation Based on Physics Metrics in Finsler ... 10Archive (2019) Special of SID Issue (Nonlinear Analysis in Engineering and Sciences), 97-114 107 hence F¯ is constant along the geodesic of F . According to

dτ = F dt, dτ¯ = F¯ dt, (7.17)

Then d log(F ) = d log(F¯). (7.18) Now,we consider F and F¯ satisfied in equation (7.18) and we apply this condition in the following calculations. Plugging the geodesic spray coeﬀicients Gr into the geodesic equation (2.1), we obtain

′ d2t B dr dt + = 0, (7.19) dτ 2 B dτ dτ The solution of equation (7.19) is dt B = 1. (7.20) dτ ′ ′ d2r B dt A dr r + ( )2 + ( )2 − F¯2 = 0, (7.21) dτ 2 2A dτ 2A dτ A With simplifying as following

′ ′ d2r B dt A dr r dr ( + ( )2 + ( )2 − F¯2 = 0) × 2A( ), (7.22) dτ 2 2A dτ 2A dτ A dτ

2 ′ ′ dr dr d r dr B dr A ( )3 + 2A − 2r dτF¯2 − = 0, (7.23) dτ dτ dτ 2 2A B2 dτ dA dr dr dr d2r d dr d dr ( )2 + A(2 ) − F¯2 (r2) − (B−1) = 0 (7.24) dr dτ dτ dτ dτ 2 dr dτ dr dτ dA dr d dr d d ( )2 + A ( )2 − F¯2 (r2) − (B−1) = 0 (7.25) dτ dτ dτ dτ dτ dτ d dr 1 [A( )2 − F¯2r2 − ] = 0, (7.26) dτ dτ B By replacing equation (7.20) in equation (7.26), we have the following equation

d dr dt [A( )2 − F¯2r2 − B( )2] = 0, (7.27) dτ dτ dτ By regarding the equation (2.1), we have

d (−2r2F¯2 − F 2) = 0, (7.28) dτ dr dF 2 − 4r F¯2 − = 0, (7.29) dτ dτ dF 2 dr = −4r F¯2, (7.30) dτ dτ And dF 2 dr dt = −4r F¯2, (7.31) dτ dt dτ www.SID.ir 108Archive of SID M. Farahmandy Motlagh, A. Behzadi

dF dr Because dτ = 0 thus dt is small. Plugging the geodesic spray coeﬀicients Gt into the geodesic equation (2.1), we gain ′ d2t B dr dt + = 0, (7.32) dt2 B dτ dτ d2t ′ 2 B dτ dτ = − dr, (7.33) dt B dτ And dt ln = − ln B, (7.34) dτ We consider 2 t t − r r − 2 2 F = By y Ay y r FFS, (7.35) 2GM 2GM −1 Where B = (1 − ) and A = (λ − ) , and the ”Finslerian sphere” FFS is of the form of λr √ r (1 − ε2 sin2 θ)yθyθ + sin2 θyφyφ ε sin2 θyφ F = − , (7.36) FS 1 − ε2 sin2 θ 1 − ε2 sin2 θ ≤ π Where 0 ε < 1. If we consider the orbit of a particle confined to the equatorial plane θ = 2 , Finslerian sphere simplifies as, dφ ε dφ dφ F = dτ − dτ = dτ , (7.37) FS 1 − ε2 1 − ε2 1 + ε

dFFS By using dτ = 0, we obtain dφ = J±, (7.38) dτ Where J± = J(1 ± ε). Plugging equation (7.38) into the equation (7.35) and consider F = 0 for massless particles we obtain 2GM dt 2GM dr r2 dφ (1 − )( )2 − (1 − )−1( )2 − ( )2 = 0, (7.39) r dτ r dτ (1 ± ε)2 dτ By using equations (7.39) and (7.38) we have dr r2 2GM dφ ( )2 − 1 = − (1 − )( )2, (7.40) dτ (1 ± ε)2 r dτ dr 1 r2 2GM ( )2 − = − (1 − ), (7.41) dφ dφ 2 (1 ± ε)2 r ( dτ ) Plugging equation (7.38) into the equation (7.41) we obtain dr 1 r2 2GM ( )2 − = − (1 − ), (7.42) dφ J 2(1 ± ε)2 (1 ± ε)2 r After simplifying equation (7.42), we gain

√ dr 1 √ = ± dφ, (7.43) G2M 2J2+1 2 − − 2 (1 ε) ( J ) (r GM) √ We consider r − GM = G2M 2J2+1 sin α and solve the equation (7.43) we obtain J √ G2M 2J 2 + 1 1 r = sin(( )φ + c) + GM, (7.44) J (1 ± ε)

www.SID.ir Solution of Vacuum Field Equation Based on Physics Metrics in Finsler ... 10Archive (2019) Special of SID Issue (Nonlinear Analysis in Engineering and Sciences), 97-114 109

8. Weyl and Douglas tensor Let F be a Finsler metric on an n-dimensional manifold M and Gi be the geodesic coeﬀicient δGi δGi ∈ ∈ − { } i j − k of F , x M and y TxM 0 , the Riemannian curvature tensor is defined by Rjk = δxk δxj . ∂Rh k h jk k (nHki+Hik)y ̸ We know Hijk = ∂yi , Hij = Hijk and Hi = n−1 provided n = 1. The Weyl tensor can be expressed, in four dimensions, as [11]

i i i (Hjk − Hkj)y + δ Hk − δ Hj W i = Ri + j k , (8.1) jk jk n + 1

we know from the ′ A 1 λ Gt = − + , (8.2) t rA2 rA2 r2 ′ −B 1 λ Gr = − + , (8.3) r rAB rA2 r2 ′ ′ ′ ′ ′ B B A B A B Gθ = Gφ = − + + ( + ), (8.4) θ φ 2AB 2rAB 2rA2 4AB A B h h h According to definition of Hijk since Hijk is independent of y therefore Hijk = 0, Hij = 0 and Hi = 0. i i Thus Wjk = Rjk. By using the equation (8.1), we obtain

∂Gt ∂Gr Rt = − t = 0,Rr = − r = 0, rt ∂r rt ∂t ∂Gθ ∂Gφ Rθ = − θ = 0,Rφ = − φ = 0, θr ∂r φr ∂r t r θ φ Thus Gt =constant, Gr =constant, Gθ =constant and Gφ =constant. i t And other Rjk are equal zero. Since Gt = α that α is constant therefore we have

′ A 1 λ − + = α, (8.5) rA2 rA2 r2 If we solve the equation (8.5) as follows

′ 1 λ A A−2 − A−1 = (αr − ), r r Therefore we have 1 c A = (λ − αr2 + )−1, (8.6) 3 r θ Since Gθ = γ that γ is constant therefore we have

′ ′ ′ ′ ′ B B A B A B − + + ( + ) = γ, (8.7) 2AB 2rAB 2rA2 4AB A B From the equation (8.5) and equation (8.7) we conclude that

′ −B 1 λ − + = β = constant, (8.8) rAB rA2 r2

www.SID.ir 110Archive of SID M. Farahmandy Motlagh, A. Behzadi

We solved equation (8.8) as follows

′ B λ 1 = −βrA + A − , B r r We use the equation (8.6) and we suppose that α = β therefore we obtain 1 c B = (λ − αr2 + ), (8.9) 3 r From equation (8.6) and equation (8.9) we conclude B = A−1, (8.10) Therefore, if W = 0 and α = β =constant we conclude that 1 c 1 c F 2 = (λ − αr2 + )ytyt − (λ − αr2 + )−1yryr − r2F¯2, (8.11) 3 r 3 r Thus F 2 is the Ads Schwarzschild Finsler metric [12]. If F 2 express as follows 2M r2 2M r2 F 2 = (λ − + )ytyt − (λ − + )−1yryr − r2F¯2, (8.12) r b2 r b2 i i 2 We conclude that Rjk = 0 and this conclude that Wjk = 0, thus F is the Ads Schwarzschild Finsler metric and F 2 is conformally flat. Now we consider Douglas tensor as follows yhG + σ i, j, k){δhG } Dh = Gh − ijk ( i jk , (8.13) ijk ijk n + 1 Where 1 ∂Gh ∂G Gi = Γi yiyj,Gh = jk ,G = Gr ,G = ij , (8.14) 2 jk ijk ∂yi ij ijr ijk ∂yk h ⇒ h ⇒ i t If tensor Douglas D = 0 thus Dijk = 0 Gijk = 0 Gi = 0. Therefore Gt = 0 and we can conclude that ′ A 1 λ − + = 0, rA2 r2A r2 Therefore we have c A = (λ + )−1, (8.15) r r Also we know that Gr = 0 therefore −B´ 1 λ − + = 0, rAB r2A r2 ′ B λA 1 = − , B r r we use the equation (8.15) and we obtain c B = (λ + ), (8.16) r From the equations (8.15) and (8.16) we conclude B = A−1, (8.17) Therefore if D = 0 we conclude that F 2 is the Schwarzschild metric.

www.SID.ir Solution of Vacuum Field Equation Based on Physics Metrics in Finsler ... 10Archive (2019) Special of SID Issue (Nonlinear Analysis in Engineering and Sciences), 97-114 111

9. The Friedmann-Lemaître-Robertson-Walker (F LRW ) solution By using an ansatz of the F LRW metric that it has the Finslerian structure dt dt a2(t) dr dr dθ dφ F 2 = − − r2a2(t)F¯2(θ, φ, ), (9.1) dτ dτ 1 − kr2 dτ dτ dτ dτ We have found an exact solution of the vacuum field equation, where geodesic spray coeﬀicients can be derived as ′ 1 2aa dr dr ′ Gt = ( + 2aa r2F¯2), (9.2) 4 1 − kr2 dτ dτ ′ a dr dt 1 kr dr dr 1 Gr = + − r(1 − kr2)F¯2, (9.3) a dτ dτ 2 1 − kr2 dτ dτ 2 ′ a dθ dt 1 dθ dr Gθ = G¯θ + + , (9.4) a dτ dτ r dτ dτ ′ a dφ dt 1 dφ dr Gφ = G¯φ + + , (9.5) a dτ dτ r dτ dτ Now, we compute Ricci scalar of equation (9.1). ∂Gµ ∂2Gµ ∂2Gµ ∂Gµ ∂Gλ RicF 2 = 2 − yλ + 2Gλ − , (9.6) ∂xµ ∂xλ∂yµ ∂yλ∂yµ ∂yλ ∂yµ Where Ric¯ denotes the Ricci scalar of the Finsler structure F¯. Since the vacuum field equation in Finsler spacetime is equivalent to the vanishing of the Ricci scalar thus ∂Gt ∂ ∂Gt ∂ ∂Gt ∂Gt ∂Gλ 2 − yλ + 2Gλ − ∂t ∂xλ ∂yt ∂yλ ∂yt ∂yλ ∂yt ′′ aa ′′ = yryr + aa r2F¯2, (9.7) 1 − kr2

∂Gr ∂ ∂Gr ∂ ∂Gr ∂Gr ∂Gλ 2 − yλ + 2Gλ − ∂r ∂xλ ∂yr ∂yλ ∂yr ∂yλ ∂yr ′′ a ′ 2 = − ytyt + r2(a + k)F¯2, (9.8) a and ∂Gθ ∂ ∂Gθ ∂ ∂Gθ ∂Gθ ∂Gλ 2 − yλ + 2Gλ − ∂θ ∂xλ ∂yθ ∂yλ ∂yθ ∂yλ ∂yθ ∂Gφ ∂ ∂Gφ ∂ ∂Gφ ∂Gφ ∂Gλ +2 − yλ + 2Gλ − ∂φ ∂xλ ∂yφ ∂yλ ∂yφ ∂yλ ∂yφ ′′ ′ 2 a a + k = Ric¯ F¯2 − 2 ytyt + 2 yryr, (9.9) a 1 − kr2 Then we rewrite equation (2.13) as

′′ ′′ ′ 2 a aa + 2a + 2k RicF 2 = −3 ytyt + yryr+ a 1 − kr2 ′′ ′ 2 (Ric¯ + r2(aa + a + k))F¯2, (9.10)

www.SID.ir 112Archive of SID M. Farahmandy Motlagh, A. Behzadi

Because of vanishing of the Ricci scalar, we have Ric¯ = λ

′′ ′′ ′ 2 a aa + 2a + 2k = 0, = 0, a 1 − kr2 ′′ ′ 2 λ + r2(aa + a + k) = 0, (9.11)

′′ a = 0 ⇒ a(t) = αt + β, (9.12)

′′ ′ 2 aa + 2a + 2k = 0 ⇒ k = −α2, (9.13) 1 − kr2 r2(α2 + k) + λ = 0, α2 + k = 0, ⇒ λ = 0, (9.14) Thus we obtain, dt dt (αt + β)2 dr dr F 2 = − − r2(αt + β)2F¯2, (9.15) dτ dτ 1 + α2r2 dτ dτ Where F¯ is Ricci flat and Ric¯ = λ. Now, we consider geodesic equation

d2xµ + 2Gµ = 0, (9.16) dτ 2

d2t + 2Gt = 0, dτ 2 2 ′ d t 2aa dr dt ′ + ( ( )2( )2 + 2aa r2F¯2) = 0, (9.17) dτ 2 1 − kr2 dt dτ

dr If, we only consider the radial motion of particles, and notice the velocity of a particle dt is small, we obtain d2t d dt = ( ) = 0, (9.18) dτ 2 dτ dτ dt = constant = A, (9.19) dτ From the another geodesic equation, we gain

d2r + 2Gr = 0, dτ 2 d2r dt ( )2 − r(1 − kr2)F¯2 = 0, (9.20) dt2 dτ Therefore d2r d2r A = 0 ⇒ = 0, (9.21) dt2 dt2 On the other hand from the equation of motion of a test particle we obtain

d2r ∂φ 2GM = − = , (9.22) dt2 ∂r r2

www.SID.ir Solution of Vacuum Field Equation Based on Physics Metrics in Finsler ... 10Archive (2019) Special of SID Issue (Nonlinear Analysis in Engineering and Sciences), 97-114 113

From the equations (9.21) and (9.22) we obtain 2GM = 0, (9.23) r2 Therefore r → ∞ or M = 0. We consider the gravitational field equation in the given Finsler spacetime [9] should be of the form

µ µ Gν = 8πF GTν , (9.24) We use the modified Einstein tensor [8] as follows 1 G ≡ Ric − g S, (9.25) µν µν 2 µν

k kl And consider Gk = g Gkl we obtain

′ 2 a 2k λ Gt = 2 + + , (9.26) t a2 a2 r2a2

′′ a λ Gr = 2 + , (9.27) r a r2a2 and ′′ ′ 2 a a + k Gθ = Gφ = 2 + , (9.28) θ φ a a2 as we know that the energy-momentum tensor to be of the form

µ − − − Tν = diag(ρ, p, p, p) (9.29) we use equations (9.24) and(9.26), obtain

t t Gt = 8πF GTt ′ 2 a 2k λ 2 + + = 8π Gρ, (9.30) a2 a2 r2a2 F According to the equations (9.12)-(9.14) we have

ρ = 0, (9.31)

Moreover

θ θ Gθ = 8πF GTθ ′′ ′ 2 a a + k 2 + = 8π G(−p), (9.32) a a2 F If we notice to the equations (9.12)-(9.14) we obtain

p = 0, (9.33)

Thus ρ and p are p = 0, ρ = 0 therefore the energy-momentum tensor is zero.

www.SID.ir 114Archive of SID M. Farahmandy Motlagh, A. Behzadi

10. conclusions In this paper, we investigated the LTB solutions for (3.1) containing dust with p = 0. We have obtained the R(t, r) and S(t, r) with considering establish a new solution of Rµν = 0. Our solutions show that two dimensional subspace F¯ has constant Ricci curvature.Then, we compute the covariant derivative Einstein tensor and show that it is not conserved in the Finsler spacetime. Moreover, we obtained the Kretschman scalar for the considering anstaz. Finally, we determined Ks singularity is at R = 0 and we showed one example. Singularity theorems have been one of the most important developments in the theory of classical general relativity . Also, we consider an ansatz of the Schwarzschild metric and we show results of vacuum solution are different from of [8].We consider Weyl tensor and we conclude that the Finsler metric is the Ads Schwarzschild and conformally flat.

References

[1] W. V Wessel Valkenburg, Complete solutions to the metric of spherically collapsing dust in an expanding spacetime with a cosmological constant, Springerlink.com, (2014). [2] J. T. Firouzjaee, R. Mansouri, Radiation from the LTB black hole, arXiv:1104.0530 [gr-qc], (2012). [3] I. Gkigkitzis, I. Haranas and O. Ragos,Kretschmann Invariant and relations between spacetime singularity entropy and information, Physics International 5 (1), 103-111, (2014). [4] A. Krasinski, C. Hellaby, Formation of a galaxy with a central blackhole in the Lemaitre-Tolman model. Phys. Rev. D. Vol. 69, 043502(2004). [5] R. G. Vishwakarma, Pramana – J. Phys. 85, 1101 (2015). [6] A. N. Makarenko and V. V. Obukhov, Entropy 14, 1140 (2012). [7] R. G. Vishwakarma, Astrophys.Space Sci. 321, 151 (2009). [8] X. Li and Z. Chang, Phys. Rev. D 90, 064049 (2014). [9] Z. Shen, Lectures on Finsler Geometry, World Scientific, Singapore, (2001). [10] K. Bolejko , M.N Célérier,Szekeres Swiss-cheese model and supernova observations,Phys. Rev. D 82, 103510, 8 November (2010). [11] P. L. Antonelli, R. S. Ingarden and M. Matsumoto, The Theory of Sprays and Finsler Spaces with Applications in Physics and Biology, Springer, Kluwer Academic, (1993). [12] M. Dafermos, A note on naked singularities and the collapse of self- gravitating Higgs fields. Adv. Theor. Math. Phys. 9, 575–591 (2005).